When is a mat foundation appropriate and how is it analyzed?

Mat foundations are used when column loads are high, allowable bearing pressure is low, columns are closely spaced, or differential settlement must be minimized. Analysis methods: rigid method (assumes mat doesn't bend, bearing pressure varies linearly), flexible method (soil modeled as springs, mat deformation computed), and finite element with soil-structure interaction. Design checks include bearing capacity, punching shear at columns, flexure (moment distribution like inverted flat slab), and differential settlement.